Underwater Robots - Gianluca Antonelli.pdf

186 7. Dynamic Control of UVMSs
Finally, the generalized force for the vehicle needed to achieve the corresponding
required force is computed as
h 0 c,0 = h 0 r,0 + U 0 1 h 1 c,1 .
7.9.1 Stability Analysis
In this Section, the stability analysis for the discussed control law isprovided.
Let consider the following scalar function
V i ( s i i , ˜ θ i )= 1
2 s i T
i M i s i i + 1
2 ˜ θ T
i K θ,i ˜ θ i . (7.73)
The scalar V i ( s i i , ˜ θ i ) > 0inview of positive definiteness of M i and K θ,i .
By differentiating V i with respect to time yields
˙V i = s i T
i M i ( ˙ν i r,i − ˙ν i i ) − ˜ θ T
i K θ,i ˙ θ ˆ
i ,
where the parameters was considered constant orslowly varying, i.e.,
˙˜θ i = − ˙ ˆ θ i .
Taking into account the equations ofmotions (7.65), and considering the
vector n i = C i ( ν i i ) ν i r,i + D i ( ν i i ) ν i r,i + g i i ( R i )itis:
˙V i = − s i T
i D i ( ν i i ) s i i + s i T �
i M i ˙ν i r,i + n i ( ν i i , ν i r,i , R i ) − h i � T
t,i − θ ˜
i K θ ˙ θ ˆ
i .
By adding and subtracting the term s i T i
i h c,i, where h i c,i is the control law as
introduced in (7.70), and by exploiting the linearity inthe parameters, the
previous equation can be rewritten as
˙V i = − s i T
i D i s i i + s i T
i
�
Y i θ i − h i t,i − Y i ˆ θ i − K v,i s i i − U i i +1
i +1h c,i+1
− ˜ θ T
i K θ,i ˙ θ ˆ
i + s i T i
i h c,i,
where Y i = Y ( R i , ν i i , ν i r,i , ˙ν i r,i )and D i = D i ( ν i i ).
By rearranging the terms one obtains:
˙V i = − s i T
i ( K v,i + D i ) s i i + s i T
i
�
Y i ˜ θ i − h i t,i + h i r,i
�
− ˜ θ T
i K θ,i ˙ ˆ θ i ,
�
+
that, taking the update law for the dynamic parameters (7.71), finally gives
˙V i = − s i T
i ( K v,i + D i ) s i i + s i T i
˜h
i t,i , (7.74)
where ˜ h i
t,i = h i r,i − h i t,i.Equation (7.74) does not have anysignificantproperty
with respect to its sign.
The Lyapunov candidate function for the UVMS is given by
V ( s 0 0 ,...,s n n , ˜ θ 0 ,..., ˜ n�
θ n )= V i ( s i i , ˜ θ i ) ,
i =0